[Paper Review] Delayed instabilities in viscoelastic solids through a metric description

Erez Y. Urbach, Efi Efrati|arXiv (Cornell University)|Nov 26, 2017

Sports Dynamics and Biomechanics3 citations

TL;DR

This paper introduces a metric-based framework for viscoelasticity that models viscoelastic solids through temporally evolving reference lengths, enabling quantitative prediction of delayed instabilities. By treating the material's stress relaxation as a dynamic reference metric, the theory predicts when and why seemingly stable viscoelastic structures—like elastomer shells—suddenly snap, with excellent agreement to experimental data on poppers and snap-through behavior.

ABSTRACT

While determining the stability of an unconstrained elastic structure is a straightforward task, this is not the case for viscoelastic structures. Seemingly elastically stable conformations of viscoelastic structures may gradually creep until stability is lost, and conversely, creeping does not necessarily imply that a structure will eventually become unstable. Understanding instabilities in viscoelastic structures requires a more intuitive description of viscoelasticity to allow analytical results and quantitative predictions. In this work we put forward a metric description of viscoelasticity in which the continua is characterized by temporally evolving reference lengths with respect to which elastic strains are measured. Formulating the three dimensional theory using metric tensors we are able to predict which structures will exhibit delayed instability due to viscoelastic flow. We also quantitatively describe the viscoelastic relaxation in free standing structures including cases where the relaxation leads to no apparent motion. We demonstrate these results and the power of the metric approach by elucidating the subtle mechanism of delayed instability in elastomer shells showing quantitative agreement with experimental measurements.

Motivation & Objective

• To address the lack of intuitive, analytical tools for predicting delayed instabilities in viscoelastic solids, which are often missed by standard constitutive models.
• To develop a physically interpretable framework that captures the slow, dissipative evolution of internal stresses without relying on memory kernels or complex simulations.
• To explain why some viscoelastic structures remain motionless despite continuous internal stress relaxation, a phenomenon that defies classical elasticity.
• To quantitatively predict the transition between stable, unstable, and metastable behaviors in thin viscoelastic shells under load.
• To validate the theory experimentally using elastomeric poppers, demonstrating precise agreement between predicted and observed snap-through times and stability phases.

Proposed method

• Introduce a metric description of viscoelasticity where the elastic response is measured relative to a time-evolving reference metric $\bar{g}_{ij}(t)$, representing the material’s internal relaxation state.
• Define the instantaneous reference length $\bar{g}_{ij}$ as a dynamic variable that evolves toward the current configuration $g_{ij}$, with a relaxation time constant governed by the material parameter $\beta$.
• Formulate the stress tensor using a modified constitutive law: $S^{ij}(t) = C^{ijkl} \left( \varepsilon_{kl}(t) + \beta \int_{-\infty}^{t} \dot{\phi}(t-s) \varepsilon_{kl}(s) ds \right)$, where $\varepsilon_{kl} = \frac{1}{2}(g_{kl} - \bar{g}_{kl})$.
• Model the viscoelastic response as a continuous creep of the reference metric $\bar{g}_{ij}$ toward the current configuration $g_{ij}$, with the system approaching a stationary state $\bar{g}^{\text{stat}} = \beta g + (1 - \beta)\bar{g}^0$.
• Apply the theory to free-standing thin shells (e.g., conical poppers), computing the energy landscape and stability based on the evolving $\bar{g}_{ij}$, enabling prediction of delayed snap-through.
• Use experimental measurements of stress relaxation to calibrate $\beta = 0.0928 \pm 0.0077$, and compare predicted stability phases with observed snap-through times in 50 different poppers.

Experimental results

Research questions

• RQ1 Why do some viscoelastic structures remain motionless despite continuous internal stress relaxation, even when seemingly unstable?
• RQ2 What determines the transition between stable, unstable, and metastable behaviors in viscoelastic shells under constant deformation?
• RQ3 How can delayed instabilities—such as snap-through in elastomer poppers—be quantitatively predicted using a physically intuitive framework?
• RQ4 To what extent does the evolving reference metric $\bar{g}_{ij}$ capture the full viscoelastic response, including stress relaxation and creep at zero load?
• RQ5 Can the metric approach predict the divergence of snap-through time near the boundary between stable and unstable regimes in thin shells?

Key findings

• The theory predicts that a viscoelastic system abruptly brought to a locally stable state may remain motionless indefinitely, even as internal stresses evolve, due to the system's reference metric $\bar{g}_{ij}$ adjusting continuously without inducing net deformation.
• The material parameter $\beta = 0.0928 \pm 0.0077$ was experimentally measured via stress relaxation, enabling quantitative prediction of stability phases in conical poppers.
• The model accurately predicts the three distinct stability phases in elastomeric poppers—stable, unstable, and metastable—based on geometric parameters $r_{\text{min}}/h$ and $r_{\text{max}}/r_{\text{min}}$.
• Experimental data on 50 different poppers show excellent agreement with theoretical phase boundaries, validating the predictive power of the metric framework.
• The theory explains the slow creeping motion preceding snap-through in poppers and Venus flytraps as a result of the slow evolution of $\bar{g}_{ij}$ toward the current configuration.
• The model reveals that delayed instability arises from frustration between the current shape and the material’s evolving stress-free state, with the reference metric $\bar{g}_{ij}$ serving as a state variable for viscoelastic history.